Abstract

Due to wide range of interest in use of bioeconomic models to gain insight into the scientific management of renewable resources like fisheries and forestry, homotopy perturbation method is employed to approximate the solution of the ratio-dependent predator-prey system with constant effort prey harvesting. The results are compared with the results obtained by Adomian decomposition method. The results show that, in new model, there are less computations needed in comparison to Adomian decomposition method.

1. Introduction

Partial
differential equations which arise in real-world physical problems are often
too complicated to be solved exactly, and even if an exact solution is
obtainable, the required calculations may be practically
too complicated, or it might be difficult to interpret the
outcome. Very recently, some promising approximate analytical solutions are
proposed such as Exp-function method, Adomian decomposition method (ADM),
variational iteration method (VIM), and homotopy perturbation method (HPM).

HPM is the most
effective and convenient method for both linear and nonlinear equations. This
method does not depend on a small parameter. Using homotopy technique in
topology, a homotopy is constructed with an embedding parameter 𝑝∈[0,1], which is considered as a “small parameter.”
HPM has been shown to effectively, easily, and accurately solve a large class
of linear and nonlinear problems with components converging to accurate
solutions. HPM was first proposed by He [1–7] and was successfully applied to
various engineering problems.

The motivation of this paper is
to extend the homotopy perturbation method (HPM) [8–17] to solve the
ratio-dependent predator-prey system. The results of HPM are compared with
those obtained by the ADM [18]. Different from ADM, where specific algorithms
are usually used to determine the Adomian polynomials, HPM handles linear and
nonlinear problems in simple manner by deforming a difficult problem into a
simple one. The HPM is useful to obtain exact and
approximate solutions of linear and nonlinear differential equations.

In this paper, we assume that
the predator in model is not of commercial importance. The prey is subjected to
constant effort harvesting with 𝑟,
a parameter that measures the effort being spent by a harvesting agency. The
harvesting activity does not affect the predator population directly. It is
obvious that the harvesting activity does reduce the predator population
indirectly by reducing the availability of the prey to the predator. Adopting a
simple logistic growth for prey population with
𝑒>0,𝑏>0,
and 𝑐>0 standing for the predator death rate, capturing rate, and
conversion rate, respectively, we formulate the problem as𝑑𝑥𝑑𝑡=𝑥(1−𝑥)−𝑏𝑥𝑦𝑦+𝑥−𝑟𝑥,𝑑𝑦=𝑑𝑡𝑐𝑥𝑦𝑦+𝑥−𝑒𝑦,(1.1) where 𝑥(𝑡) and 𝑦(𝑡) represent the fractions of population
densities for prey and predator at time 𝑡,
respectively. Equations (1.1) are to be solved according to biologically
meaningful initial conditions 𝑥(0)≥0 and 𝑦(0)≥0 [18].

2. Applications

In this section, we will apply the HPM to nonlinear differential system of
ratio-dependant predator-prey,𝐻=𝐿𝜈𝑢𝜈,𝑝1−𝑝−𝐿0𝐴𝜈𝑟+𝑝−𝑓=0,𝑝∈0,1,𝑟𝜀Ω,(2.1)where 𝐴(𝜈) is a general differential operator which can be divided into a linear part 𝐿(𝜈) and a nonlinear part 𝑁(𝜈) and 𝑓(𝑟) is a known analytical function. 𝑝∈[0,1] is an embedding
parameter, while 𝑢0 is an initial
approximation of the equation which should be solved, and satisfies the
boundary conditions.

According to the HPM (relation
(2.1)), we can construct a homotopy of system as follows: 𝜈1−𝑝2̇𝜈1+𝜈1̇𝜈1−̇𝑥0y0−̇𝑥0𝑥0𝜈+𝑝2̇𝜈1+𝜈1̇𝜈1−𝜈1−𝑏−𝑟1𝜈2+𝜈2𝜈21−𝜈1−𝑟21+𝜈31×𝜈=0,1−𝑝2̇𝜈2+𝜈1̇𝑣2−̇y0y0−x0̇y0𝜈+𝑝2̇𝜈2+𝜈1̇𝜈2+𝜈𝑒−𝑐1𝜈2+e𝜈22=0,(2.2)
where dot denotes differentiation with respect to 𝑡, and the
initial approximations are as follows:𝑣1,0(𝑡)=𝑥0𝑣(𝑡)=𝑥(0),2,0(𝑡)=𝑦0(𝑡)=𝑦(0).(2.3) Assume that the solution of (2.2)
can be written as a power series in 𝑝 as follows:𝜈1=𝜈1,0+𝑝𝜈1,1+𝑝2𝜈1,2+𝑝3𝜈1,3𝜈+⋯,2=𝜈2,0+𝑝𝜈2,1+𝑝2𝜈2,2+𝑝3𝜈2,3+⋯,(2.4) where 𝜈𝑖,𝑗(𝑖,𝑗=1,2,3,…) are functions yet to be determined. Substituting
(2.3) and (2.4) into (2.2), and arranging the coefficients of p powers, we have𝑣2,0̇𝑣1,0+𝑣1,0̇𝑣1,0+𝑣31,0−𝑣21,0+𝑣1,0̇𝑣1,1+𝑣2,0̇𝑣1,1+𝑟𝑣1,0𝑣2,0+𝑏𝑣1,0𝑣2,0−𝑣1,0𝑣2,0+𝑣2,0𝑣21,0+𝑟𝑣21,0𝑝+𝑣1,1̇𝑣1,1+𝑣1,0̇𝑣1,2+𝑣2,0̇𝑣1,2+𝑣2,1̇𝑣1,1+2𝑟𝑣1,0𝑣1,1+𝑏𝑣1,0𝑣2,1+2𝑣2,0𝑣1,0𝑣1,1+𝑟𝑣1,1𝑣2,0+𝑟𝑣1,0𝑣2,1+𝑏𝑣1,1𝑣2,0−𝑣1,0𝑣2,1−𝑣1,1𝑣2,0+𝑣2,1𝑣21,0−2𝑣1,0𝑣1,1+3𝑣21,0𝑣1,1𝑝2+𝑣1,1̇𝑣1,2+𝑣1,2̇𝑣1,1+𝑣1,0̇𝑣1,3+𝑣2,1̇𝑣1,2+𝑣2,0̇𝑣1,3+𝑣2,2̇𝑣1,1+𝑣2,0𝑣21,1−𝑣1,0𝑣2,2−𝑣1,2𝑣2,0−𝑣1,1𝑣2,1+𝑣2,2𝑣21,0+𝑟𝑣21,1+3𝑣1,0𝑣21,1−𝑣21,1+𝑏𝑣1,1𝑣2,1+𝑏𝑣1,0𝑣2,2+𝑏𝑣1,2𝑣2,0+𝑟𝑣1,0𝑣2,2+𝑟𝑣1,1𝑣2,1+𝑟𝑣1,2𝑣2,0+2𝑣2,0𝑣1,0𝑣1,2+2𝑟𝑣1,0𝑣1,2+2𝑣2,1𝑣1,0𝑣1,1+3𝑣21,0𝑣1,2−2𝑣1,0𝑣1,2𝑝3𝑣+⋯=0,2,0̇𝑣2,0+𝑣1,0̇𝑣2,0+𝑒𝑣1,0𝑣2,0−𝑐𝑣1,0𝑣2,0+𝑣2,0̇𝑣2,1+𝑣1,0̇𝑣2,1+𝑒𝑣22,0𝑝+𝑣2,1̇𝑣2,1+𝑒𝑣1,0𝑣2,1−𝑐𝑣1,0𝑣2,1+𝑒𝑣1,1𝑣2,0−𝑐𝑣1,1𝑣2,0+2𝑒𝑣2,0𝑣2,1+𝑣2,0̇𝑣2,2+𝑣1,1̇𝑣2,1+𝑣1,0̇𝑣2,2𝑝2+𝑒𝑣22,1+𝑣2,1̇𝑣2,2+𝑣2,2̇𝑣2,1+𝑣2,0̇𝑣2,3+𝑣1,1̇𝑣2,2+𝑣1,2̇𝑣2,1+𝑣1,0̇𝑣2,3+𝑒𝑣1,0𝑣2,2+𝑒𝑣1,1𝑣2,1−𝑐𝑣1,0𝑣2,2−𝑐𝑣1,1𝑣2,1+𝑒𝑣1,2𝑣2,0−𝑐𝑣1,2𝑣2,0+2𝑒𝑣2,0𝑣2,2𝑝3+⋯=0.(2.5) In order to obtain the
unknown of
𝜈𝑖,𝑗(𝑥,𝑡),𝑖,𝑗=1,2,3,…,
we must construct and solve the following system which includes 6 equations,
considering the initial conditions of 𝜈𝑖,𝑗(0)=0,𝑖,𝑗=1,2,3,… :𝑣2,0̇𝑣1,0+𝑣1,0̇𝑣1,0𝑣=0,31,0−𝑣21,0+𝑣1,0̇𝑣1,1+𝑣2,0̇𝑣1,1+𝑣1,0𝑣2,0+𝑏𝑣1,0𝑣2,0−𝑣1,0𝑣2,0+𝑣2,0𝑣21,0+𝑟𝑣21,0𝑣=0,1,1̇𝑣1,1+𝑣1,0̇𝑣1,2+𝑣2,0̇𝑣1,2+𝑣2,1̇𝑣1,1+2𝑟𝑣1,0𝑣1,1+𝑏𝑣1,0𝑣2,1+2𝑣2,0𝑣1,0𝑣1,1+𝑟𝑣1,1𝑣2,0+𝑟𝑣1,0𝑣2,1+𝑏𝑣1,1𝑣2,0−𝑣1,0𝑣2,1−𝑣1,1𝑣2,0+𝑣2,1𝑣21,0−2𝑣1,0𝑣1,1+3𝑣21,0𝑣1,1𝑣=0,2,0̇𝑣2,0+𝑣1,0̇𝑣2,0=0,𝑒𝑣1,0𝑣2,0−𝑐𝑣1,0𝑣2,0+𝑣2,0̇𝑣2,1+𝑣1,0̇𝑣2,1+𝑒𝑣22,0𝑣=0,2,1̇𝑣2,1+𝑒𝑣1,0𝑣2,1−𝑐𝑣1,0𝑣2,1+𝑒𝑣1,1𝑣2,0−𝑐𝑣1,1𝑣2,0+2𝑒𝑣2,0𝑣2,1+𝑣2,0̇𝑣2,2+𝑣1,1̇𝑣2,1+𝑣1,0̇𝑣2,2=0.(2.6) From (2.4), if the first three approximations are sufficient, then setting 𝑝=1 yields the approximate solution of (1.1)
to𝑥(𝑡)=lim𝑝→1𝑣1(𝑡)=𝑘=3𝑘=0𝑣1,𝑘(𝑡),𝑦(𝑡)=lim𝑝→1𝑣2(𝑡)=𝑘=3𝑘=0𝑣2,𝑘(𝑡).(2.7) Therefore,v1,0(𝑡)=𝑥0𝑣(𝑡)=𝑥(0),(2.8)1,1x(𝑡)=−0𝑥20−𝑥0−𝑦0+𝑥0𝑦0+𝑟𝑦0+𝑏𝑦0+𝑟𝑥0𝑡𝑥0+𝑦0,𝑣(2.9)1,21(𝑡)=2𝑥0+𝑦03𝑥0𝑡23𝑦0𝑥20−𝑥20𝑏𝑦0+2𝑥30𝑏𝑦0+3𝑥40𝑟+6𝑥30𝑦20−3𝑦30𝑥0+𝑥30𝑟2−9𝑥30𝑦0+6𝑥40𝑦0−9𝑥20𝑦20+2𝑦30𝑥20−2𝑥30𝑟−2𝑟𝑦30−2𝑏𝑦30+𝑏2𝑦30+𝑟2𝑦30+𝑥20𝑏𝑦0𝑟+3𝑥0𝑟𝑦20𝑏+𝑦20𝑥0𝑒𝑏+𝑏𝑥20𝑦0𝑒−𝑏𝑥20𝑦0𝑐−3𝑥0𝑏𝑦20+3𝑥0𝑦20+3𝑦30𝑥0𝑟+3𝑦30𝑥0𝑏−6𝑥20𝑟𝑦0+2𝑥50−3𝑥40+𝑦30+2𝑟𝑦30𝑏+9𝑥30𝑟𝑦0−6𝑥0𝑟𝑦20+9𝑥20𝑦20𝑟+5𝑥20𝑦20𝑏+𝑥30+3𝑥0𝑟2𝑦20+3𝑥20𝑟2𝑦0,𝑣(2.10)2,0(𝑡)=𝑦0𝑣(𝑡)=𝑦(0),(2.11)2,1y(𝑡)=0−𝑒𝑥0+𝑐𝑥0−𝑒𝑦0𝑡𝑦0+𝑥0,𝑣(2.12)2,21(𝑡)=−2𝑦0+𝑥03𝑦0𝑡23𝑦0𝑒𝑥20𝑐+𝑦20𝑐𝑥0𝑒+2𝑒𝑥30𝑐−𝑐𝑥20𝑦0−𝑐𝑥0𝑦20−𝑐2𝑥30+𝑐𝑥30𝑦0+𝑐𝑥20𝑦0𝑟+𝑐𝑥0𝑦20𝑏+𝑐𝑥20𝑦20+𝑐𝑥0𝑦20𝑟−𝑒2𝑥30−3𝑦0𝑒2𝑥20−3𝑦20𝑒2𝑥0−𝑦30𝑒2.(2.13) We also obtained 𝑣1,3 and 𝑣2,3,
but because they were too long to maintain, we skip them and only use them in
the final numerical results. In this manner, the other components can be easily
obtained by substituting (2.8)
through (2.13) into (2.7) as follows:x𝑥(𝑡)=𝑥(0)−0𝑥20−𝑥0−𝑦0+𝑥0𝑦0+𝑟𝑦0+𝑏𝑦0+𝑟𝑥0𝑡𝑥0+𝑦0+12𝑥0+𝑦03𝑥0𝑡2(3𝑦0𝑥20−𝑥20𝑏𝑦0+2𝑥30𝑏𝑦0+3𝑥40𝑟+6𝑥30𝑦20−3𝑦30𝑥0+𝑥30𝑟2−9𝑥30𝑦0+6𝑥40𝑦0−9𝑥20𝑦20+2𝑦30𝑥20−2𝑥30𝑟−2𝑟𝑦30−2𝑏𝑦30+𝑏2𝑦30+𝑟2𝑦30+𝑥20𝑏𝑦0𝑟+3𝑥0𝑟𝑦20𝑏+𝑦20𝑥0𝑒𝑏+𝑏𝑥20𝑦0𝑒−𝑏𝑥20𝑦0𝑐−3𝑥0𝑏𝑦20+3𝑥0𝑦20+3𝑦30𝑥0𝑟+3𝑦30𝑥0𝑏−6𝑥20𝑟𝑦0+2𝑥50−3𝑥40+𝑦30+2𝑟𝑦30𝑏+9𝑥30𝑟𝑦0−6𝑥0𝑟𝑦20+9𝑥20𝑦20𝑟+5𝑥20𝑦20𝑏+𝑥30+3𝑥0𝑟2𝑦20+3𝑥20𝑟2𝑦0+𝑣1,3y⋯,𝑦(𝑡)=𝑦(0)+0−𝑒𝑥0+𝑐𝑥0−𝑒𝑦0𝑡𝑦0+𝑥0−12𝑦0+𝑥03×𝑦0𝑡23𝑦0𝑒𝑥20𝑐+𝑦20𝑐𝑥0𝑒+2𝑒𝑥30𝑐−𝑐𝑥20𝑦0−𝑐𝑥0𝑦20−𝑐2𝑥30+𝑐𝑥30𝑦0+𝑐𝑥20𝑦0𝑟+𝑐𝑥0𝑦20𝑏+𝑐𝑥20𝑦20+𝑐𝑥0𝑦20𝑟−𝑒2𝑥30−3𝑦0𝑒2𝑥20−3𝑦20𝑒2𝑥0−𝑦30𝑒2+𝑣2,3⋯.(2.14)

3. Numerical Results and Comparison with ADM

For comparison with the results obtained by ADM [18],
the parameter values
in four
cases are considered in Table 1.

Table 1: Parameter values used for illustration
purposes.

Results of four terms approximation for 𝑥(𝑡),𝑦(𝑡) obtained by using HPM and ADM [18] are
presented in (3.1), respectively: Case1∶𝑥≈0.5−0.35𝑡+0.19476𝑡2−0.107288𝑡3,𝑦≈0.3−0.1125𝑡+0.018808𝑡2−0.0011284𝑡3,Case2∶𝑥≈0.5+0.05𝑡+0.012265𝑡2−0.0016032𝑡3,𝑦≈0.3−0.1125𝑡+0.024433𝑡2−0.00398199𝑡3,Case3∶𝑥≈0.3+0.0799t+0.00533t2−0.00115𝑡3,𝑦≈0.6−0.08𝑡+0.01866𝑡2−0.00231𝑡3,Case4∶𝑥≈0.5+0.07857𝑡−0.016020𝑡2−0.00119873𝑡3,𝑦≈0.2+0.051428𝑡+0.0055918𝑡2+0.00002245𝑡3,Case1∶𝑥≈0.5−0.35000𝑡+0.19476𝑡2−0.10728𝑡3,𝑦≈0.3−0.11250𝑡+0.018809𝑡2−0.0011286𝑡3,Case2∶𝑥≈0.5+0.05000𝑡+0.012266𝑡2−0.0016034𝑡3,𝑦≈0.3−0.11250𝑡+0.024434𝑡2−0.0039821𝑡3,Case3∶𝑥≈0.3+0.08000t+0.005333t2−0.0011555𝑡3,𝑦≈0.6−0.08000𝑡+0.018667𝑡2−0.0023112𝑡3,Case4∶𝑥≈0.5+0.07857𝑡−0.016021𝑡2−0.0011984𝑡3,𝑦≈0.2+0.051430𝑡+0.0055920𝑡2+0.00002246𝑡3.(3.1) Figures 1–4 show the
relations between prey and predator populations versus time.

A noteworthy observation from Figure 1 is that prey and predator species can become extinct simultaneously for some
values of parameters, regardless of the initial values. Thus, overexploitation
of the prey population by constant effort harvesting process together with high
predator capturing rate may lead to mutual extinction as a possible outcome of
predator-pray interaction. In Figure 2, only the predator population gradually
decreases and becomes extinct despite the availability of increasing prey
population. This can be attributed to the effect of the predator death rate,
being greater than the conversion rate and low constant prey harvesting as
shown in Case 2 (see Table 1). Figures
3 and 4 illustrate the possibility of
predator and prey long-term coexistence. Depending on the initial values, both
prey and predator populations increase or reduce in order to allow long-term
coexistence [18].

4. Conclusion

Homotopy perturbation method was employed to
approximate the solution of the ratio-dependent predator-prey system with
constant effort prey harvesting. The results obtained here were compared with
results of Adomian decomposition method. The results show that there is less
computations needed in comparison to ADM.